I'm a senior in a college calculus class. The following problem was
assigned, but the next day, after no one had completed it, our professor
attempted to do it but quit after 25 minutes. Problem: Find the equation
of the line tangent to the ellipse b^2*x^2 + a^2*y^2 = a^2*b^2 in the
first quadrant that forms with the coordinate axes the triangle of
smallest possible area (a & b are positive constants).

If F represents the focus of the ellipse and P is the position of the
starship Enterprise on the ellipse... you must fire the engines when the
ratio vector T/vector N is equal to the eccentricity of the ellipse.

I need to calculate the area and center of gravity of a crescent: take a
10.00" diameter circle center at x,y 0,0 and add a 9.00" diameter circle
center at x,y 2,0. The crescent is formed in 2 areas that are not
overlapping.

In the surgical connection of a small artery to an existing larger
artery, attention must be paid to minimising the viscous resistance
to the blood flow... Investigate the distance from B of the join and
hence the angle of attachment that minimises the resistance to flow.

Problem: y = 4-x2 ; x axis - a) Draw a figure showing the region and a
rectangular element of area; b) express the area of the region as the
limit of a Riemann sum; c) find the limit in part b by evaluating a
definite integral by the second fundamental theorem of the calculus.

I learned that dy/dx was a notation that implied 'derivative of y with
respect to x.' But in some calculus work, dy and dx seem to have
individual meanings and dy/dx is treated as a fraction. How is this
possible?

Points A and B represent pressure sensors in fixed positions on the
base of a round tank. The chord through CD represents the water level
in the tank. Lines a and b are the heights of water registered by each
sensor...

In the plane x + y + 3z = 7, my book says we find the closest point to
the origin by [d/[n]^2] * n. I don't understand WHY this operation
gives us the closest point. I'd appreciate it if you can help explain
how it works.